If f is smooth function and p n 1 is polynomial of

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Iffis smooth function, andpn-1is polynomial of degree atmostn-1interpolatingfatnpointst1, . . . , tn, thenf(t)-pn-1(t) =f(n)()n!(t-t1)(t-t2)· · ·(t-tn)whereis some (unknown) point in interval[t1, tn]Since pointis unknown, this result is not particularlyuseful unless bound on appropriate derivative offisknownMichael T. HeathScientific Computing33 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMonomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and ConvergenceInterpolating Continuous Functions, continuedIf|f(n)(t)|Mfor allt2[t1, tn], andh= max{ti+1-ti:i= 1, . . . , n-1}, thenmaxt2[t1,tn]|f(t)-pn-1(t)|Mhn4nError diminishes with increasingnand decreasingh, butonly if|f(n)(t)|does not grow too rapidly withn< interactive example >Michael T. HeathScientific Computing34 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMonomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and ConvergenceHigh-Degree Polynomial InterpolationInterpolating polynomials of high degree are expensive todetermine and evaluateIn some bases, coefficients of polynomial may be poorlydetermined due to ill-conditioning of linear system to besolvedHigh-degree polynomial necessarily has lots of “wiggles,”which may bear no relation to data to be fitPolynomial passes through required data points, but it mayoscillate wildly between data pointsMichael T. HeathScientific Computing35 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMonomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and ConvergenceConvergencePolynomial interpolating continuous function may notconverge to function as number of data points andpolynomial degree increasesEqually spaced interpolation points often yieldunsatisfactory results near ends of intervalIf points are bunched near ends of interval, moresatisfactory results are likely to be obtained withpolynomial interpolationUse of Chebyshev points distributes error evenly andyields convergence throughout interval for any sufficientlysmooth functionMichael T. HeathScientific Computing36 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMonomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and ConvergenceExample: Runge’s FunctionPolynomial interpolants of Runge’s function atequallyspacedpointsdo notconverge< interactive example >Michael T. HeathScientific Computing37 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMonomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and ConvergenceExample: Runge’s FunctionPolynomial interpolants of Runge’s function atChebyshevpointsdoconverge< interactive example >Michael T. HeathScientific Computing38 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMonomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and Convergence

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